Download Text S1 Snitkin and Segrè, Epistatic interaction maps relative to

Text S1
Snitkin and Segrè, Epistatic interaction maps relative to multiple metabolic phenotypes
Supplementary Methods
1. Determination of mutant fluxes using flux balance models
In order to utilize the flux balance analysis framework to study epistasis with respect to
reaction flux phenotypes, it is critical that flux predictions in deletion mutants be as
accurate as possible. To this end, we relied on a recent evaluation of the abilities of
different objective functions to predict fluxes in single gene deletion mutants [1]. In the
aforementioned study, 8 different objective functions were evaluated based on the
correlations of predicted fluxes for 13 single gene deletion mutants, with corresponding
experimental flux measurements. It was found that the best performing method was an
experimentally constrained implementation of the previously described Minimization Of
Metabolic Adjustment (MOMA) algorithm [2]. This approach selects the set of mutant
fluxes that has the minimal Manhattan distance from an experimentally constrained wildtype flux solution. This experimentally constrained wild-type solution is computed by
adding constraints to limit the values of any flux for which an experimental measurement
is available. Specifically, the range of each of these fluxes is constrained to be up to a
fraction  larger or smaller than the corresponding measured flux value, with additional
slack being allowed to account for the error in the experimental measurement. This
approach for adding experimental constraint to a flux balance optimization, which has
been reported before[3], can be formalized as follows:
m
min  | viW T |
i 1
s.t. S  vW T  0
 j  vWj T   j
m  (m *  )   m  vmW T  m  (m *  )   m
where v W T is the wild-type flux solution,  j (  j ) are the lower (upper) bounds on flux ,
λm is the experimental measurement of flux m and εm is the associated error in the
measurement. Here, following [3], we use =0.1. Given the experimentally constrained
wild-type flux solution v W T , a mutant flux solution can be identified by solving the
 

following linear programming MOMA problem:
m
min

| v
KO
i
 v Wi T |
i1
s.t. S  v KO  0
 j  v KO
j
j
v giKO  0

where v KO are the predicted fluxes for the deletion mutant g and v giKO are the fluxes
constrained to zero based on the assumption that gene g is required for the flux to be
active. A comparison of model predictions using this approach and corresponding
experimental measurements is shown in Figure S1. Experimental fluxes used to constrain
 the model and assess model predictions were taken from
Blank et al. [4].
2. Different metrics for quantifying epistasis
Our assessment of different metrics for quantifying epistasis consists of the evaluation of
two previously used metrics and one novel metric. The two previously used metrics
(multiplicative and additive), along with others, were recently evaluated using
experimentally determined single and double deletion mutant growth rates [5]. The
authors found that the multiplicative performed similarly to the additive definition, but
varied when applied to mutants with large growth defects. Here, we explore the behavior
of these metrics with respect to metabolic flux phenotypes. Below is a brief description of
the previously used metrics, in addition to our novel Z-score metric.
For all metric definitions let the phenotype under consideration be some flux v. The wildtype, two single deletion mutants and double mutant fluxes are represented as vWT, vx, vy
and vxy, respectively. For application of the additive and multiplicative definitions all
fluxes were normalized by the wild-type flux, with the corresponding normalized mutant
phenotypes being represented as Wx, Wy and Wxy. Finally, the normalized phenotypic
effects of the single and double mutants are δx, δy and δxy, respectively. All these
definitions are formalized as follows:
Wx 
vx
 1 x
vW T
Multiplicative
The multiplicative definition assumes that, in absence of epistasis, the normalized
phenotype of the double mutant is equal to the product of the normalized phenotypes of
the single mutants. This definition is based on the assumption that each mutation causes
the phenotype to change by some fraction.
  (1   xy )  (1   x )(1   y )  Wxy  WxWy
Additive
The additive definition assumes that the phenotypic effect of two independent mutations
combine additively in a double mutant. This definition is based on the assumption that
each mutation causes the phenotype to change by some fixed amount.
  (1 xy )  (1 x  y )  W xy  (W x  W y 1)
Z score

The goal of the Z score metric ( Z ), which was inspired by the previously described S
metric [6], was to provide an unbiased approach for quantifying epistasis that does not
rely on any assumptions as to how the phenotypic effects of independent mutations
should combine. We felt that such a metric would be ideal for the current multi there is no evidence that the effects of mutations should
phenotype analysis, where
combine in the same way for all phenotypes.
We avoid making assumptions about how phenotypic effects of mutations should
combine by quantifying phenotypic effects of mutations relative to a large set of other
mutations, instead of in absolute terms. The assumption that we then make is that if
mutation A and mutation B do not interact, then the relative effect of mutation A, as
compared to all other mutations, should be the same in the absence or presence of
mutation B. The justification for this approach is the assumption that each mutation will
interact with only a minority of other mutations. Therefore, this majority of noninteracting mutations act as a standard by which to judge which mutations are in fact
interacting (See Table S1 for illustration of this concept).
The way in which we quantify the relative effects of mutations is by Z-score normalizing
the distributions of mutant phenotypes separately for the wild-type, and also each mutant
background (hence the name of the method). Epistasis is then quantified for a pair of
mutants x and y, as follows:
Z  Z(W x ,W )  Z(W xy ,Wy )
Where a “blank” as subscript in W indicates the set of all genes, Z(Wx ,W ) is the Z score
for the phenotypic effect of deleting gene x, relative to the distribution of all other single

mutant effects and Z(W xy ,Wy ) is the Z score for the phenotypic effect of deleting genes x
and y, relative to the distribution of all of other double mutants involving the deletion of

y. Note that for a given gene deletion pair, there are two possible Z scores:

Z  Z(W x ,W )  Z(W xy ,Wy )
and


Z  Z(W y ,W )  Z(W xy ,W x )
For all analyses, the mean of these two scores was used. If Z is close to zero, then this
indicates that the phenotypic effect of a mutation, relative to all other mutations, is the

same in the presence or absence of the second mutation. On the other hand, if Z is large,
then this is evidence that the phenotypic effect of a mutation is in fact dependant on the

presence or absence of the second mutation.
3. Evaluation of different metrics for quantifying epistasis 
To assess performance of the three different metrics, we evaluated the congruence
between any two genes (i.e. the similarity in how they interact with all other genes), and
asked how well this congruence correlates with gene function similarity, across all gene
pairs. Previous work has shown that genes with similar patterns of epistatic interactions,
tend to reside in the same pathways [7, 8]. Therefore we expected that a more accurate
quantification of epistasis would result in a stronger relationship between genetic
congruence and common pathway membership. A comparison of the performance of
different metrics averaged across all phenotypes shows that while the performance is
similar for different metrics across all phenotypes (Figure S2A), it can vary greatly, for a
given metric, across different phenotypes (Figure S2B). This is an indication that
epistasis seems overall robustly detectable independent of the metric used, but that
different phenotypes are differently informative as epistasis indicators. For all analyses in
the main text we used a multiplicative definition, and validated that results were
consistent with other metrics.
4. Delineation of different classes of epistasis
In previous studies using flux balance analysis to study epistasis only biomass production
rate was utilized as a phenotype. Furthermore, in most of these studies, wild-type and
mutant fluxes were determined by assuming maximal biomass production. Because
mutations in flux balance models are implemented by adding constraints to represent the
effect of the gene deletion, the biomass production of the mutant can never be predicted
to be greater than in the wild-type, and hence no gene deletion can have a beneficial
effect. In other words, the mutant optimization problem is the same as the wild-type, but
with additional constraints, thus dictating that that the maximum value cannot be greater
for the mutant. This limitation on the behavior of the mutant phenotype limited the
classes of epistasis that could be observed to two classes (See Figure S3A).
This limitation on the behavior of the flux phenotypes does not hold here because we are
considering non-growth phenotypes, and additionally we are not assuming optimal
growth in the wild-type nor the mutants (See above). The flexibility in the behavior of
flux phenotypes, in both the wild-type and mutant backgrounds, results in an expansion
of the classes of epistasis we can observe. We delineated the different classes of epistasis
based on the guidelines and terminology described in a supplemental reference [9]. The
way in which these eight different classes of epistasis can occur is illustrated in Figure
S3B and S3C.
For the purpose of analyses performed in the main manuscript it would have been
needlessly convoluted to work with all 8 possible classes of epistasis; therefore we
consolidated them into 2 classes, synergistic and antagonistic. While, technically,
diminishing returns, compensatory and super-compensatory are all sub-classes of
antagonistic epistasis, we decided to group super-compensatory interactions in the
synergistic class. The reason for this is that we found that interactions, which were supercompensatory for one phenotype, were often synergistic for another. Furthermore, both of
these interaction classes are suggestive of major flux reroutings. The decision to classify
super-compensatory interactions as synergistic did not impact the results of our analysis,
as these represent relatively few interactions.
Left out of this Figure S3 is an additional class of epistasis, which does not fit into
standard classification guidelines. Specifically, there are some instances in which a flux
increases in response to one gene deletion, decreases in response to a second and where
flux in the double mutant does not fit the multiplicative expectation. Such interactions are
not obviously synergistic or antagonistic. For this reason, we did not include these
interactions in any analyses based on partitioning the interactions into synergistic or
antagonistic classes.
Supplementary Discussion
1. Coverage of the 3D epistatic interaction map
In order to quantify the increase in interaction coverage achieved by monitoring all flux
phenotypes the cumulative interaction coverage was determined for all flux phenotypes,
sorted by the number of unique interactions contributed (Figure 4). For this analysis
interactions with respect to each phenotype were counted only if their multiplicative
values were greater than one standard deviation from the mean epistasis found with
respect to the given phenotype. We found trends similar to those seen in Figure 4 when
different standard deviation cutoffs were applied (data not shown). Overall, it can be seen
in Figure 4 that ~80 phenotypes are required to capture all unique interactions, indicating
the unique information provided by different phenotypes. The metabolic processes with
which these ~80 phenotypes are annotated mirrors the distribution of process annotations
for all 293 phenotypes, indicating that a sampling of all metabolic functions is required to
identify all interactions (Figure S6). Importantly, despite the fact that a large proportion
of phenotypes are required to identify all interactions, the majority of interactions can be
identified with far less, suggesting that intelligent phenotype selection can be performed
to maximize interaction coverage. Furthermore, the functional specificity of different
phenotypes suggests that phenotypes can be selected to maximize insight into metabolic
pathways of interest.
2. Flux rerouting underlying specific interactions with respect to metabolic
phenotypes
Model flux predictions relevant to the following examples were analyzed with the help of
the VisANT network visualization tool (http://visant.bu.edu) [10]. For each of the three
interactions below, we provide an explanation of the underlying biochemistry, along with
metabolic network images showing the most relevant flux changes observed.
Interaction between glutamate metabolism (glt1) and respiratory chain (cox1), with
respect to glycerol secretion
This interaction was described also in the main text. A detailed analysis of mutant fluxes
(Figure S7) revealed that the basis for the synergistic interaction between glt1 and cox1,
with respect to glycerol secretion, is the involvement of both the genes and the phenotype
in NADH/NAD balancing. Specifically, glt1 is an NADH dependent reaction, and in its
absence, glutamate is synthesized by the NADPH dependant glutamate dehydrogenase
reaction (Figure S7B). Therefore, the deletion of glt1 leaves an excess of NADH, which
is in turn oxidized via the respiratory chain (Figure S7C). In the cox1 mutant, the
respiratory chain is no longer functional, and the redox imbalance is alleviated through
glycerol production (Figure S7D). Hence, there is an unexpectedly large increase in
glycerol secretion in the double mutant, relative to what is observed in the single mutants.
Interactions among phospholipid biosynthetic genes with respect to inositol uptake
Examination of the specific interactions among phospholipid biosynthetic genes with
respect to inositol uptake revealed that they were all antagonistic interactions among
genes involved in the Kennedy pathway. The Kennedy pathway is one of the two
pathways by which yeast can synthesize the phospholipids, phosphatidylcholine (PC) and
phosphatidylethanolamine (PE). Deletion of any of the Kennedy pathway genes results in
a rerouting of flux through phospholipid biosynthetic pathways, such that PC and PE can
be produced through the alternative route. Associated with this rerouting is an increased
production of the phospholipid phosphatidylinositol (PI) (Figures S8B and S8C). PI
requires inositol for its synthesis, thereby explaining why deletion of Kennedy pathway
genes leads to an increase in inositol uptake. The reason why the interactions among the
genes are antagonistic is that any of the deletions is sufficient to disable the Kennedy
pathway, meaning that additional deletions have no further phenotypic effect.
Interactions between serine biosynthesis (ser1) and TCA cycle (sdh1) with respect to
succinate secretion
The ser1 encoded enzyme catalyzes an essential step in the biosynthesis of serine from
the glycolytic intermediate 3-phosphoglycerate (Figure S9A). In the absence of ser1,
serine is predicted to be synthesized from a pathway originating from glyoxylate (Figure
S9B). In order to produce the glyoxylate necessary for the synthesis of serine, the
glyoxylate cycle is utilized. In addition to producing glyoxylate, the cycle also results in
production of succinate, which is partially consumed by the sdh1 catalyzed succinate
dehydrogenase reaction. Therefore, in the absence of sdh1, the extra succinate produced
in the ser1 mutant is predicted to be secreted, accounting for the synergistic interaction
found with respect to succinate secretion (Figures S9C and S9D).
3. Interactions which are synergistic with respect to one phenotype and
antagonistic with respect to another
In the main text we reported the observation that gene deletions often interact
synergistically with respect to some phenotypes, and antagonistically with respect to
others. On first thought, this observation is counterintuitive, as the class of epistasis
observed between two gene deletions is thought to depend on the functional relationship
between two genes. We hypothesize that this observation can be explained by the fact
that a gene deletion can have multiple, potentially unrelated, impacts on the metabolic
network, which in turn each prompt flux changes in the network. Therefore, depending
on the flux phenotype being observed, a different class of interaction can be observed,
depending on the two deletions impact on that particular phenotype. Here, we provide an
example, observed in the current data set, to crystallize this hypothesis.
Interaction between glutamate metabolism (glt1) and respiratory chain (cox1), is
synergistic with respect to glycerol secretion and antagonistic with respect to acetate
secretion
Above, we described a synergistic interaction observed between the deletions of glt1 and
cox1, with respect to glycerol secretion. We found that this interaction was a consequence
of glutamate metabolism, the electron transport chain and glycerol production, all playing
a role in oxidizing NADH, and hence maintaining proper redox balance in the cell.
Examining the interactions between glt1 and cox1 with respect to other phenotypes
revealed an antagonistic interaction with respect to acetate secretion.
How can glt1 and cox1 interact synergistically with respect to one phenotype, and
antagonistically with respect to another? The reason is that while the loss of either glt1 or
cox1 has the same impact on the NADH/NAD balance in the cell (e.g. leads to excess
NADH), the impact of their loss on the NADPH/NADP balance differs. Specifically,
when glt1 is deleted, glutamate biosynthesis proceeds through the NADPH dependent
glutamate dehydrogenase reaction, thereby creating a need for additional NADPH.
Deletion of cox1 on the other hand, does not affect NADPH balance.
In yeast, production of acetate through the ald6-catalyzed reaction is associated with
reduction of NADP to NADPH. Therefore, the differential effects of the deletions of cox1
and glt1 on NADPH usage are manifested through changes in acetate secretion. Upon
deletion of cox1, acetate production is decreased, as flux is rerouted from acetate
production, to ethanol production, in order to oxidize excess NADH. In the double
deletion of glt1 and cox1, acetate production increases relative to what it is in the cox1
single mutant, as there is a greater need for NADPH in the absence of glt1. This
observation is by definition an antagonistic interaction, whereby the deletion of cox1
results in less of a reduction in acetate secretion, in the presence of the glt1 deletion.
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